# Learning ground states of gapped quantum Hamiltonians with Kernel Methods

arXiv: https://arxiv.org/abs/2303.08902 · HTML: https://mlsanity.com/paper/2303.08902/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 68 references that are in the map.
- First posted on arXiv 15 Mar 2023.

## Builds on (most cited first, 8 of 8)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [The Noisy Power Method: A Meta Algorithm with Applications](https://mlsanity.com/paper/1311.2495/) (2013)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)
- [Scalable neural quantum states architecture for quantum chemistry](https://mlsanity.com/paper/2208.05637/) (2022)
- [Infinite Neural Network Quantum States: Entanglement and Training Dynamics](https://mlsanity.com/paper/2112.00723/) (2021)
- [Complex-Valued Kernel Methods for Regression](https://mlsanity.com/paper/1610.09915/) (2016)

## Built on by (most cited first, 2 of 2)

- [Hybrid Ground-State Quantum Algorithms based on Neural Schr"odinger Forging](https://mlsanity.com/paper/2307.02633/) (2023)
- [Exploring the Effect of Basis Rotation on NQS Performance](https://mlsanity.com/paper/2512.17893/) (2025)

## Abstract (arXiv)

Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using kernel ridge regression, we provide numerical evidence that the learning assumption is verified by applying our scheme to find the ground states of several prototypical interacting many-body quantum systems, both in one and two dimensions, showing the flexibility of our approach.